# Smallest graphs achieving the Stinson bound

## TL;DR

This paper constructs smaller graphs that achieve the maximum possible information ratio for perfect secret sharing schemes, improving previous results by significantly reducing the number of vertices needed.

## Contribution

It introduces a new family of graphs with fewer vertices that still attain the optimal information ratio, advancing the efficiency of secret sharing schemes.

## Key findings

- Constructed smaller graphs with optimal information ratio
- Achieved the same ratio on exponentially fewer vertices
- Improved bounds from previous constructions

## Abstract

Perfect secret sharing scheme is a method of distribute a secret information $s$ among participants such that only predefined coalitions, called qualified subsets of the participants can recover the secret, whereas any other coalitions, the unqualified subsets cannot determine anything about the secret. The most important property is the efficiency of the system, which is measured by the information ratio. It can be shown that for graphs the information ratio is at most $(\delta+1)/2$ where $\delta$ is the maximal degree of the graph. Blundo et al. constructed a family of $\delta$-regular graphs with information ratio $(\delta+1)/2$ on at least $c\cdot 6^\delta$ vertices. We improve this result by constructing a significantly smaller graph family on $c\cdot 2^\delta$ vertices achieving the same upper bound both in the worst and the average case.